At its core, algebra is like placing your coat on the
movie seat next to you while your friend goes to get popcorn. People
looking at that seat don’t actually see your friend. What they see
is a placeholder meant to signify that a person is sitting there.
If we were to make an equation to represent this, it would be:

coat = your friend

or simply:

c = friend

That’s what algebra is all about. Letters (variables)
are used to represent an undefined quantity in an expression or equation.
So in the equation 3w + 6 = 23, the
variable w is merely a placeholder for some real
number that has yet to be determined.

Typically, you need to get to the bottom of the mystery
and determine the numeric value of w. This is called manipulating
the equation byisolating the variable.
It sounds a bit wicked, but you won’t get in trouble for doing it. Isolating
the variable means you want to mess with the equation until
you are left with only the variable w on one side
of the equation. An actual number should be on the other side of
the equals sign. To manipulate the equation, you can add, subtract,
multiply, divide, square, or take the square root of numbers, but
whatever you do to one side of the equation, you must also do the
other side. You can be manipulative, but you have to be fair about
it.

The starting point:

3w + 6 = 23

subtracting 6 from both sides:

3w + 6 – 6 = 23 – 6

3w = 17

dividing 3 from both sides:

Ta da! The variable has been isolated, and by manipulating
the equation you get an actual numeric value for it.

A majority of the algebra items you see on the SAT will
require some bit of manipulation. We can bring back our sample item
from the Anatomy section and manipulate it to find the answer.

4.

If and , what does b equal?

Start with the equation. Here we want to isolate b:

Now we plug in the given value for a.

Then we square both sides.

4096 = b

This item brings up a good point. The equation with the
variable w is pretty straightforward. Most real
SAT items aren’t. They are filled with fractions, exponents, and k,
where k is a variable representing the kitchen
sink. It’s all still manipulation, though, so remember to do to
one side what you do to the other, and it will all work out.

Distribution and Factoring

These are two nifty little equation-manipulating gimmicks
that crop up on algebra items from time to time. Distribution takes
a term outside a set of parentheses and distributes it across all
the terms inside the parentheses. So if you have 6(3w +
8), you could distribute the 6 in the following manner:

6(3w + 8) = (6)(3w)
+ (6)(8) = 18w + 48

“Great, but so what?” you might say in your most bored,
underwhelmed voice. Well, it just so happens that many really complicated-looking
algebra items become simpler after distribution. But don’t
take our word for it:

By distributing, you now have two similar terms that cancel
each other out:

Factoring is distribution in reverse. With
factoring, you notice common factors that can be taken out of an
equation. Starting with , you can take
out the greatest common factor, 7rb, so that the
factored equation becomes .